Disintegration theorem

In mathematics, the disintegration theorem is a result in measure theory and probability theory. It rigorously defines the idea of a non-trivial "restriction" of a measure to a measure zero subset of the measure space in question. It is related to the existence of conditional probability measures. In a sense, "disintegration" is the opposite process to the construction of a product measure. Consider the unit square in the Euclidean plane , . Consider the probability measure defined on by the restriction of two-dimensional Lebesgue measure to . That is, the probability of an event is simply the area of . We assume is a measurable subset of . Consider a one-dimensional subset of such as the line segment . has -measure zero; every subset of is a -null set; since the Lebesgue measure space is a complete measure space, While true, this is somewhat unsatisfying. It would be nice to say that "restricted to" is the one-dimensional Lebesgue measure , rather than the zero measure. The probability of a "two-dimensional" event could then be obtained as an integral of the one-dimensional probabilities of the vertical "slices" : more formally, if denotes one-dimensional Lebesgue measure on , then for any "nice" . The disintegration theorem makes this argument rigorous in the context of measures on metric spaces. (Hereafter, will denote the collection of Borel probability measures on a topological space .) The assumptions of the theorem are as follows: Let and be two Radon spaces (i.e. a topological space such that every Borel probability measure on is inner regular e.g. separable metric spaces on which every probability measure is a Radon measure). Let . Let be a Borel-measurable function. Here one should think of as a function to "disintegrate" , in the sense of partitioning into . For example, for the motivating example above, one can define , , which gives that , a slice we want to capture. Let be the pushforward measure . This measure provides the distribution of (which corresponds to the events ).